Johanna N. Y. Franklin

Degrees of Categoricity and the Hyperarithmetic Hierarchy

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of Fokina, Kalimullin, and R. Miller to show that for every computable ordinal α, 0 is the degree of categoricity of some computable structure A. We show additionally that for α a computable successor ordinal, every degree 2-c.e. in and above 0 is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees of categoricity is Π1 complete.

Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets

We show that if a point in a computable probability space X satisfies the ergodic recurrence property for a computable measure-preserving T : X → X with respect to effectively closed sets, then it also satisfies Birkhoff’s ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff’s ergodic theorem for the shift operator with respect to Π1 classes. This answers a question of Hoyrup and Rojas. Several theorems in ergodic theory state that almost all points in a probability space behave in a regular fashion with respect to an ergodic transformation of the space. For example, if T : X → X is ergodic, then almost all points in X recur in a set of positive measure: Theorem 1 (See [5]). Let (X,μ) be a probability space, and let T : X → X be ergodic. For all E ⊆ X of positive measure, for almost all x ∈ X, T(x) ∈ E for infinitely many n. Recent investigations in the area of algorithmic randomness relate the hierarchy of notions of randomness to the satisfaction of computable instances of ergodic theorems. This has been inspired by Kučera’s classic result characterising MartinLöf randomness in the Cantor space. We reformulate Kučera’s result using the general terminology of [4]. Definition 2. Let (X,μ) be a probability space, and let T : X → X be a function. Let C be a collection of measurable subsets of X. A point x ∈ X is a Poincaré point for T with respect to C if for all E ∈ C of positive measure for infinitely many n, T(x) ∈ E. The Cantor space 2 is equipped with the fair-coin product measure λ. The shift operator σ on the Cantor space is the function σ(a0a1a2 . . . ) = a1a2 . . . . The shift operator is ergodic on (2, λ). Received by the editors July 20, 2010 and, in revised form, April 5, 2011 and April 8, 2011. 2010 Mathematics Subject Classification. Primary 03D22; Secondary 28D05, 37A30. The second author was partially supported by the Marsden Grant of New Zealand. The third author was supported by the National Science Foundation under grants DMS0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness. 1Recall that if (X,μ) is a probability space, then a measurable map T : X → X is measure preserving if for all measurable A ⊆ X, μ ( T−1A ) = μ(A). We say that a measurable set A ⊆ X is invariant under a map T : X → X if T−1A = A (up to a null set). A measure-preserving map T : X → X is ergodic if every T -invariant measurable subset of X is either null or conull. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication 3623 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3624 J.N.Y. FRANKLIN, N. GREENBERG, J.S. MILLER, AND K.M. NG Theorem 3 (Kučera [7]). A sequence R ∈ 2 is Martin-Löf random if and only if it is a Poincaré point for the shift operator with respect to the collection of effectively closed (i.e., Π1) subsets of 2 . Building on work of Bienvenu, Day, Mezhirov and Shen [2], Bienvenu, Hoyrup and Shen generalised Kučera’s result to arbitrary computable ergodic transformations of computable probability spaces. Theorem 4 (Bienvenu, Hoyrup and Shen [3]). Let (X,μ) be a computable probability space, and let T : X → X be a computable ergodic transformation. A point x ∈ X is Martin-Löf random if and only if it is a Poincaré point for T with respect to the collection of effectively closed subsets of X. One of the most fundamental regularity theorems is due to Birkhoff (see [5]). Birkhoff’s Ergodic Theorem. Let (X,μ) be a probability space, and let T : X → X be ergodic. Let f ∈ L(X). Then for almost all x ∈ X,

LOWNESS AND HIGHNESS PROPERTIES FOR RANDOMNESS NOTIONS

Given two relativizable classes R and P and a real A, we say that A is in Low(R,P) if R P A and that A is in High(R,P) if R A P. In this paper, we survey the current results on highness and lowness for Kurtz, Schnorr, recursive, Martin-Lof, and weak 2-randomness.

Van Lambalgen's Theorem and High Degrees

We show that van Lambalgen’s Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen’s Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen’s Theorem holds with respect to recursive randomness.

Degrees that Are Low for Isomorphism

We say that a degree is low for isomorphism if, whenever it can compute an isomorphism between a pair of computable structures, there is already a computable isomorphism between them. We show that while there is no clear-cut relationship between this property and other properties related to computational weakness, the low-forisomorphism degrees contain all Cohen 2-generics and are disjoint from the Martin-Lof randoms. We also consider lowness for isomorphism with respect to the class of linear orders.

LOWNESS FOR DIFFERENCE TESTS

We show that being low for dierence tests is the same as being computable and therefore lowness for dierence tests is not the same as lowness for dierence randomness. This is the rst known example of a randomness notion where lowness for the randomness notion and lowness for the test notion do not coincide. Additionally, we show that for every incomputable set A, there is a dierence test T A relative to A which cannot even be covered by nitely many unrelativized dierence tests.

Local Computability for Ordinals

We examine the extent to which well orders satisfy the properties of local computability, which measure how effectively the finite suborders of the ordinal can be presented. Known results prove that all computable ordinals are perfectly locally computable, whereas \(\omega_1^\mathrm{CK}\) and larger countable ordinals are not. We show that perfect local computability also fails for uncountable ordinals, and that ordinals \(\alpha\geq \omega_1^\mathrm{CK}\) are θ-extensionally locally computable for all \(\theta \omega_1^\mathrm{CK}\), nor when \(\theta=\omega_1^\mathrm{CK}\leq\alpha<\omega_1^\mathrm{CK}\cdot\omega\).

Subclasses of the Weakly Random Reals

The weakly random reals contain not only the Schnorr random reals as a subclass but also the weakly 1-generic reals and therefore the n-generic reals for every n. While the class of Schnorr random reals does not overlap with any of these classes of generic reals, their degrees may. In this paper, we describe the extent to which this is possible for the Turing, weak truth-table, and truth-table degrees and then extend our analysis to the Schnorr random and hyperimmune reals.

Algorithmic Randomness and Fourier Analysis

Suppose 1 < p < ∞. Carleson’s Theorem states that the Fourier series of any function in Lp[−π, π] converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every f ∈ Lp[−π, π] given natural computability conditions on f and p.

Strength and Weakness in Computable Structure Theory

We survey the current results about degrees of categoricity and the degrees that are low for isomorphism as well as the proof techniques used in the constructions of elements of each of these classes. We conclude with an analysis of these classes, what we may deduce about them given the sorts of proof techniques used in each case, and a discussion of future lines of inquiry.